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Empirical Economics

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02-11-2020

Managing power supply interruptions: a bottom-up spatial (frontier) model with an application to a Spanish electricity network

Authors: Pablo Argüelles, Luis Orea

Published in: Empirical Economics | Issue 6/2021

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Abstract

In December 2013, a new electricity law was approved in Spain as part of an electricity market reform including a new remuneration scheme for distribution companies. This remuneration scheme was updated in December 2019, and the new regulatory framework introduced a series of relevant modifications that aim to encourage regulated firms to reduce their power supply interruptions using a benchmarking approach. While some managerial decisions can prevent electricity power supply interruptions, other managerial decisions are more oriented toward mitigating the consequences of these interruptions. This paper examines the second type of decision using a unique dataset referring to the power supply interruptions of a Spanish distribution company network between 2013 and 2019. We focus our analysis on the effect which grid automatization has on restoration times, the relative efficiency of the maintenance staff, and the importance of its location. We estimate a bottom-up spatial model and a stochastic frontier model to examine both external and internal power supply interruptions at municipal level. While our frontier model is standard, our spatial model differs from a conventional one in that it is developed from scratch using the information of each individual power supply interruption.

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Analyzing the frequency of PSI is also more challenging as we should cope with an excessive number of zero values in our data. The development and estimation of zero-inflated econometric models in non-frontier settings has become widespread. See Yang et al. (2017) for a comparison of methods. Our models, however, allow examining whether the number of outages matters when estimating the coefficients of interest.

The time lag between outages usually lasts milliseconds. Therefore, we cannot affirm that one outage impacts on two municipalities simultaneously from an engineering point of view, even though as humans we perceive that both PSI have started simultaneously.

In this sense, our model looks like a multilevel or hierarchical SAR model, which is becoming increasingly popular in social sciences. See Corrado and Fingleton (2016) for a summary of these models. In these models it is assumed that there exist a number of well-defined groups organized within a hierarchical structure, such as classes within schools. Much of the multilevel literature assumes that inter-individual interaction is restricted to within group boundaries. From a spatial perspective, this implies that the inter-individual interactions are restricted spatially in a similar fashion to that of our bottom-up spatial model.

Notice that the distance between municipalities does not matter in our ‘electricity’ application. So, it does not make sense to compute a spatial weight matrix using an inverse function of the distance which is common in conventional spatial models.

Notice that Eq. (4) cannot be estimated in a conventional spatial econometric application because only, say, the whole GDP in one region is observed and not the portion of such GDP that actually depends on the GDP of neighboring regions.

Equation (4) can also be estimated using maximum likelihood (ML) techniques if we assume that \( \omega_{mt} \) follows a normal distribution, i.e., \( \omega_{mt} \sim N\left( {0,\sigma_{\omega } = e^{{\tau_{0} }} } \right) \).

In this sense, Skevas (2020) shows that the endogeneity issues that exist in conventional SAR models such as \( Y_{t} = {{\gamma \varOmega }}Y_{t} + \omega_{t} \) are caused by the fact that the dependent variable of any individual appears both on the left and right-hand side of the spatial autoregressive model, after replacing \( Y_{t} \) on the right-hand side of the above equation with \( Y_{t} = {{\gamma \varOmega }}Y_{t} + \omega_{t} \). A time–space recursive model such as \( Y_{t} = {{\gamma \varOmega }}Y_{t - 1} + \omega_{t} \) overcomes such a bias because for a particular observation, while \( Y_{t} \) appears on the left-hand-side, the right-hand-side contains \( Y_{t - 2} \) after replacing \( Y_{t - 1} \) with \( Y_{t - 1} = {{\gamma \varOmega }}Y_{t - 2} + \omega_{t - 1} \), and the endogeneity issue is wiped out.

Notice that the total duration of external PSI in (4) can decomposed into the number of external PSI that affect municipality \( m \) in period \( t \) (\( J_{mt} \)) and the average duration of these interruptions (\( \bar{e}_{mt} = \mathop \sum \nolimits_{j = 1}^{{J_{mt}}} e_{jmt}/J_{mt} \)). That is, \( YE_{mt} = J_{mt} \cdot \bar{e}_{mt} \). This decomposition suggests that using a per outage specification of (4) allows us to focus on outages’ duration (i.e., on \( \bar{e}_{mt} \)) and not on the number of such outages (i.e., on \( J_{mt} \)).

If \( \tau_{0} \) is allowed to be different from zero, we obtain the so-called RSCFG-μ model. This model nests the original RSCFG model in which \( \tau_{0} = 0 \) is imposed and therefore it assumes that \( u_{mt} \) follows a half-normal distribution.

The regulator distinguishes between urban areas (municipalities with more than 20,000 supplies, including provincial capitals, even if they do not reach the previous figure), semi-urban areas (municipalities with supplies between 2000 and 20,000, excluding provincial capitals), and rural areas (municipalities with less than 2000 suppliers).

More than 3000 grid segments that cross the border between two geographically adjacent municipalities have been identified. We identified the municipalities involved in each interconnection, the number of connections, capacities (kVA), sections (mm2), voltage (kV), location (overhead/underground) of all of them.

For each one, we have identified the name, identification code, municipality name where they are installed, municipality identification code, UMTS (latitudinal and longitudinal location), rated capacity (kVA), and outdoor, indoor or underground location.

We have identified more than 400 thousand grid segments. For all of them their number (#), capacity (kVA), section (mm²), voltage (kV) and location (overhead/underground) of the interconnection have also been identified.

The reference failure type is a mixture of failures that cannot be included in the other three categories.

Recall that we are using a power-adjusted measure of the restoration times that depends on the capacity affected by the outages.

See Table 4 and Fig. 1 as well.

In both cases we have used spatial lags of \( Y_{mt} \). Larger biases are expected if the standard SAR models are estimated using \( YE_{mt} \), i.e., using a wrong spatially dependent variable.

Obviously, another network feature that determines the number of outages is the network length. In the limit case where a network is lacking (i.e., NL = 0), the number of outages is zero as there is no equipment exists which can be damaged by any failure.

They are available upon request to the authors.

At the bottom of this table we provide the correlation coefficients for several efficiency scores. We find a very large correlation (about 99%) between the efficiency scores of our general specification (9) and the efficiency scores obtained using the two nested specifications (7) and (8).

We can also reject the original RSCFG model in which \( \tau_{0} \) = 0 in favor of the more general RSCFG model in which \( \tau_{0} \) is allowed to be different from zero.

If the lack of accessibility has been extraordinarily relevant for occasional outages, its effect should be captured by the corresponding inefficiency score. As is pointed out later on, this seems to happen in Huesca province.

The lack of significance in the logged specification seems to indicate that \( DIGT \) is larger in municipalities that have more outages, this tending to offset the faster restoration of each outage.

For instance, while 95% of the network in Huesca province is made up with overhead lines and its population is widely scattered over a wide area, the overhead lines in Zaragoza province only represent 35% of the network in Zaragoza province due a much more urbanized area.

Similar comments can be made if we split the above two groups of observations into four groups.

Asturias has a very dispersed population requiring supply, except in the central area of Asturias where the urban population is concentrated. In rural areas, due to the challenging orography, there are long 132 kV lines that running as backbones through the valleys which are later transformed into 20 kV or 6 kV for more local distribution. The failure of one of these long 132 kV lines can have great relevance for the TIEPI because, although they affect fewer supply points than for the case of the central region, the repair time can be much longer.

Huesca has a distribution network similar to the one existing in rural areas of the Asturian network, both in terms of orography and their constructive solution, mainly aerial and with a 132 kV trunk line. The failure of one of these long 132 kV lines can again have a great effect on our restoration times.

Madrid, like Valencia and Alicante, has a distribution network connected to the transport network through 220/20 kV transformers, distributed in concentrated urban areas and with a high degree of underground network, which can prove more difficult to repair.

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Title: Managing power supply interruptions: a bottom-up spatial (frontier) model with an application to a Spanish electricity network
Authors: Pablo Argüelles
Luis Orea
Publication date: 02-11-2020
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 6/2021
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-020-01968-3

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